Complete Guide To Corporate Finance
Bonds  Bond Valuation
The fundamental principle of bond valuation is that the bond's value is equal to the present value of its expected (future) cash flows. The valuation process involves the following three steps:
1. Estimate the expected cash flows.
2. Determine the appropriate interest rate or interest rates that should be used to discount the cash flows.
3. Calculate the present value of the expected cash flows found in step one by using the interest rate or interest rates determined in step two.
Determining Appropriate Interest Rates
The minimum interest rate that an investor should accept is the yield for a riskfree bond (a Treasury bond for aU.S. investor). The Treasury security that is most often used is the ontherun issue because it reflects the latest yields and is the most liquid.
For nonTreasury bonds, such as corporate bonds, the rate or yield that would be required would be the ontherun government security rate plus a premium that accounts for the additional risks that come with nonTreasury bonds.
As for the maturity, an investor could just use the final maturity date of the issue compared to the Treasury security. However, because each cash flow is unique in its timing, it would be better to use the maturity that matches each of the individual cash flows.
Computing a Bond's Value
First, we need to find the present value (PV) of the bond's future cash flows. The present value is the amount that would have to be invested today to generate that future cash flow. PV is dependent on the timing of the cash flow and the interest rate used to calculate the present value. To figure out the value, the PV of each individual cash flow must be found. Then, just add the figures together to determine the bond's price.
After you calculate the expected cash flows, you will need to add the individual cash flows:
Let's throw some numbers around to further illustrate this concept.
Example: The Value of a Bond
Bond GHJ matures in five years with a coupon rate of 7% and a maturity value of $1,000. For simplicity's sake, let's assume that the bond pays annually and the discount rate is 5%.
The cash flow for each of the years is as follows:
Year One = $70
Year Two = $70
Year Three = $70
Year Four = $70
Year Five = $1,070
Thus, the PV of the cash flows is as follows:
Year One = $70 / (1.05) to the 1^{st} power = $66.67
Year Two = $70 / (1.05) to the 2^{nd} power = $ 63.49
Year Three = $70 / (1.05) to the 3^{rd} power = $ 60.47
Year Four = $70 / (1.05) to the 4^{th} power = $ 57.59
Year Five = $1,070 / (1.05) to the 5^{th} power = $ 838.37
Now to find the value of the bond:
Value = $66.67 + $63.49 + $60.47 + $57.59 + $838.37
Value = $1,086.59
How Does the Value of a Bond Change?
As rates increase or decrease, the discount rate that is used also changes. Let's change the discount rate in the above example to 10% to see how it affects the bond's value.
Example: The Value of a Bond when Discount Rates Change
PV of the cash flows is:
Year One = $70 / (1.10) to the 1^{st} power = $ 63.63
Year Two = $70 / (1.10) to the 2^{nd} power = $ 57.85
Year Three = $70 / (1.10) to the 3^{rd} power = $ 52.63
Year Four = $70 / (1.10) to the 4^{th} power = $ 47.81
Year Five = $1,070 / (1.10) to the 5^{th} power = $ 664.60
Value = 63.63 + 57.85 + 52.63 + 47.81 + 664.60 = $ 886.52
How Does a Bond's Price Change as it Approaches its Maturity Date?
As a bond moves closer to its maturity date, its price will move closer to par. There are three possible scenarios:
1.If a bond is at a premium, the price will decline over time toward its par value.
2. If a bond is at a discount, the price will increase over time toward its par value.
3. If a bond is at par, its price will remain the same.
To show how this works, let's use our original example of the 7% bond, but now let's assume that a year has passed and the discount rate remains the same at 5%.
Example: Price Changes Over Time
Let's compute the new value to see how the price moves closer to par. You should also be able to see how the amount by which the bond price changes is attributed to it being closer to its maturity date.
PV of the cash flows is:
Year One = $70 / (1.05) to the 1^{st} power = $66.67
Year Two = $70 / (1.05) to the 2^{nd} power = $ 63.49
Year Three = $70 / (1.05) to the 3^{rd} power = $ 60.47
Year Four = $1,070 / (1.05) to the 4^{th} power = $880.29
Value = $66.67 + $63.49 + $60.47 + $880.29 = $1,070.92
As the price of the bond decreases, it moves closer to its par value. The amount of change attributed to the year's difference is $15.67.
An individual can also decompose the change that results when a bond approaches its maturity date and the discount rate changes. This is accomplished by first taking the net change in the price that reflects the change in maturity, then adding it to the change in the discount rate. The two figures should equal the overall change in the bond's price.
Computing the Value of a Zerocoupon Bond
A zerocoupon bond may be the easiest of securities to value because there is only one cash flow  the maturity value.
The formula to calculate the value of a zero coupon bond that matures N years from now is as follows:
Example: The Value of a ZeroCoupon Bond
For illustration purposes, let's look at a zero coupon with a maturity of three years and a maturity value of $1,000 discounted at 7%.
I = 0.035 (.07 / 2)
N = 3
Value of a ZeroCoupon Bond
= $1,000 / (1.035) to the 6^{th} power (3*2)
= $1,000 / 1.229255
= $813.50
Arbitragefree Valuation Approach
Under a traditional approach to valuing a bond, it is typical to view the security as a single package of cash flows, discounting the entire issue with one discount rate. Under the arbitragefree valuation approach, the issue is instead viewed as various zerocoupon bonds that should be valued individually and added together to determine value. The reason this is the correct way to value a bond is that it does not allow a riskfree profit to be generated by "stripping" the security and selling the parts at a higher price than purchasing the security in the market.
As an example, a fiveyear bond that pays semiannual interest would have 11 separate cash flows and would be valued using the appropriate yield on the curve that matches its maturity. So the markets implement this approach by determining the theoretical rate the U.S. Treasury would have to pay on a zerocoupon treasury for each maturity. The investor then determines the value of all the different payments using the theoretical rate and adds them together. This zerocoupon rate is the Treasury spot rate. The value of the bond based on the spot rates is the arbitragefree value.
Determining Whether a Bond Is Under or Over Valued
What you need to be able to do is value a bond like we have done before using the more traditional method of applying one discount rate to the security. The twist here, however, is that instead of using one rate, you will use whatever rate the spot curve has that coordinates with the proper maturity. You will then add the values up as you did previously to get the value of the bond.
You will then be given a market price to compare to the value that you derived from your work. If the market price is above your figure, then the bond is undervalued and you should buy the issue. If the market price is below your price, then the bond is overvalued and you should sell the issue.
How Bond Coupon Rates and Market Rates Affect Bond Price
If a bond's coupon rate is above the yield required by the market, the bond will trade above its par value or at a premium. This will occur because investors will be willing to pay a higher price to achieve the additional yield. As investors continue to buy the bond, the yield will decrease until it reaches market equilibrium. Remember that as yields decrease, bond prices rise.
1. Estimate the expected cash flows.
2. Determine the appropriate interest rate or interest rates that should be used to discount the cash flows.
3. Calculate the present value of the expected cash flows found in step one by using the interest rate or interest rates determined in step two.
Determining Appropriate Interest Rates
The minimum interest rate that an investor should accept is the yield for a riskfree bond (a Treasury bond for a
For nonTreasury bonds, such as corporate bonds, the rate or yield that would be required would be the ontherun government security rate plus a premium that accounts for the additional risks that come with nonTreasury bonds.
As for the maturity, an investor could just use the final maturity date of the issue compared to the Treasury security. However, because each cash flow is unique in its timing, it would be better to use the maturity that matches each of the individual cash flows.
Computing a Bond's Value
First, we need to find the present value (PV) of the bond's future cash flows. The present value is the amount that would have to be invested today to generate that future cash flow. PV is dependent on the timing of the cash flow and the interest rate used to calculate the present value. To figure out the value, the PV of each individual cash flow must be found. Then, just add the figures together to determine the bond's price.
PV at time T = expected cash flows in period T / (1 + I) to the T power 
After you calculate the expected cash flows, you will need to add the individual cash flows:
Value = present value @ T1 + present value @ T2 + present value @T_{n} 
Let's throw some numbers around to further illustrate this concept.
Example: The Value of a Bond
Bond GHJ matures in five years with a coupon rate of 7% and a maturity value of $1,000. For simplicity's sake, let's assume that the bond pays annually and the discount rate is 5%.
The cash flow for each of the years is as follows:
Year One = $70
Year Two = $70
Year Three = $70
Year Four = $70
Year Five = $1,070
Thus, the PV of the cash flows is as follows:
Year One = $70 / (1.05) to the 1^{st} power = $66.67
Year Two = $70 / (1.05) to the 2^{nd} power = $ 63.49
Year Three = $70 / (1.05) to the 3^{rd} power = $ 60.47
Year Four = $70 / (1.05) to the 4^{th} power = $ 57.59
Year Five = $1,070 / (1.05) to the 5^{th} power = $ 838.37
Now to find the value of the bond:
Value = $66.67 + $63.49 + $60.47 + $57.59 + $838.37
Value = $1,086.59
How Does the Value of a Bond Change?
As rates increase or decrease, the discount rate that is used also changes. Let's change the discount rate in the above example to 10% to see how it affects the bond's value.
Example: The Value of a Bond when Discount Rates Change
PV of the cash flows is:
Year One = $70 / (1.10) to the 1^{st} power = $ 63.63
Year Two = $70 / (1.10) to the 2^{nd} power = $ 57.85
Year Three = $70 / (1.10) to the 3^{rd} power = $ 52.63
Year Four = $70 / (1.10) to the 4^{th} power = $ 47.81
Year Five = $1,070 / (1.10) to the 5^{th} power = $ 664.60
Value = 63.63 + 57.85 + 52.63 + 47.81 + 664.60 = $ 886.52
 As we can see from the above examples, an important property of PV is that for a given discount rate, the older a cash flow value is, the lower its present value.
 We can also compute the change in value from an increase in the discount rate used in our example. The change = $1,086.59  $886.52 = $200.07.
 Another property of PV is that the higher the discount rate, the lower the value of a bond; the lower the discount rate, the higher the value of the bond.

How Does a Bond's Price Change as it Approaches its Maturity Date?
As a bond moves closer to its maturity date, its price will move closer to par. There are three possible scenarios:
1.If a bond is at a premium, the price will decline over time toward its par value.
2. If a bond is at a discount, the price will increase over time toward its par value.
3. If a bond is at par, its price will remain the same.
To show how this works, let's use our original example of the 7% bond, but now let's assume that a year has passed and the discount rate remains the same at 5%.
Example: Price Changes Over Time
Let's compute the new value to see how the price moves closer to par. You should also be able to see how the amount by which the bond price changes is attributed to it being closer to its maturity date.
PV of the cash flows is:
Year One = $70 / (1.05) to the 1^{st} power = $66.67
Year Two = $70 / (1.05) to the 2^{nd} power = $ 63.49
Year Three = $70 / (1.05) to the 3^{rd} power = $ 60.47
Year Four = $1,070 / (1.05) to the 4^{th} power = $880.29
Value = $66.67 + $63.49 + $60.47 + $880.29 = $1,070.92
As the price of the bond decreases, it moves closer to its par value. The amount of change attributed to the year's difference is $15.67.
An individual can also decompose the change that results when a bond approaches its maturity date and the discount rate changes. This is accomplished by first taking the net change in the price that reflects the change in maturity, then adding it to the change in the discount rate. The two figures should equal the overall change in the bond's price.
Computing the Value of a Zerocoupon Bond
A zerocoupon bond may be the easiest of securities to value because there is only one cash flow  the maturity value.
The formula to calculate the value of a zero coupon bond that matures N years from now is as follows:
Maturity value / (1 + I) to the power of the number of years * 2 Where I is the semiannual discount rate. 
Example: The Value of a ZeroCoupon Bond
For illustration purposes, let's look at a zero coupon with a maturity of three years and a maturity value of $1,000 discounted at 7%.
I = 0.035 (.07 / 2)
N = 3
Value of a ZeroCoupon Bond
= $1,000 / (1.035) to the 6^{th} power (3*2)
= $1,000 / 1.229255
= $813.50
Arbitragefree Valuation Approach
Under a traditional approach to valuing a bond, it is typical to view the security as a single package of cash flows, discounting the entire issue with one discount rate. Under the arbitragefree valuation approach, the issue is instead viewed as various zerocoupon bonds that should be valued individually and added together to determine value. The reason this is the correct way to value a bond is that it does not allow a riskfree profit to be generated by "stripping" the security and selling the parts at a higher price than purchasing the security in the market.
As an example, a fiveyear bond that pays semiannual interest would have 11 separate cash flows and would be valued using the appropriate yield on the curve that matches its maturity. So the markets implement this approach by determining the theoretical rate the U.S. Treasury would have to pay on a zerocoupon treasury for each maturity. The investor then determines the value of all the different payments using the theoretical rate and adds them together. This zerocoupon rate is the Treasury spot rate. The value of the bond based on the spot rates is the arbitragefree value.
Determining Whether a Bond Is Under or Over Valued
What you need to be able to do is value a bond like we have done before using the more traditional method of applying one discount rate to the security. The twist here, however, is that instead of using one rate, you will use whatever rate the spot curve has that coordinates with the proper maturity. You will then add the values up as you did previously to get the value of the bond.
You will then be given a market price to compare to the value that you derived from your work. If the market price is above your figure, then the bond is undervalued and you should buy the issue. If the market price is below your price, then the bond is overvalued and you should sell the issue.
How Bond Coupon Rates and Market Rates Affect Bond Price
If a bond's coupon rate is above the yield required by the market, the bond will trade above its par value or at a premium. This will occur because investors will be willing to pay a higher price to achieve the additional yield. As investors continue to buy the bond, the yield will decrease until it reaches market equilibrium. Remember that as yields decrease, bond prices rise.
 If a bond's coupon rate is below the yield required by the market, the bond will trade below its par value or at a discount. This happens because investors will not buy this bond at par when other issues are offering higher coupon rates, so yields will have to increase, which means the bond price will drop to induce investors to purchase these bonds. Remember that as yields increase, bond prices fall.